阈值博弈：最大化期望收益

Maximize Your Gain

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

A nonnegative rv U has DF F and density $\mathbf{f} = \mathbf{F}^{\prime}$ ; its mean $\mu$ and variance o2 are both finite.A game is offered, as follows: you may choose a nonnegative number c; if $\mathrm{U} > \mathrm{c}$ then you win the amount c, otherwise you win nothing.

As an example, suppose U is the height (measured in cm) of the next person entering a specific public train station.If you choose $\mathrm{c} = 100$ then you will almost surely win that amount.A value of $\mathrm{c} = 200$ would double your amount if you win, but of course drastically reduce your winning probability.

a.Find an equation to characterize the value of c that maximizes the expected gain.

b.Give a characterization of the optimal value of c in terms of the hazard function of U (see page 2 for the definition of the hazard function) .

c.Derive c explicitly for an exponential rv with rate $\lambda$ (see page 1 for a definition) .How large is the maximum expected gain?

解析

选择阈值 $c\\ge 0$ ，若 $U>c$ 得到 $c$ ，否则 0。期望收益\n\n $\nG(c)=c\\,[1-F(c)].\n$ \n\n一阶条件\n\n $\nG'(c)=1-F(c)-c f(c)=0\\Rightarrow \\boxed{1-F(c)=c f(c)}.\n$ \n\n若 $U\\sim\\mathrm{Exp}(\\lambda)$ ，则 $1-F(c)=e^{-\\lambda c}$ 、 $f(c)=\\lambda e^{-\\lambda c}$ ，得到 $\\lambda c=1$ ，所以\n\n $\n\\boxed{c^*=\\frac{1}{\\lambda}},\\qquad \\boxed{G(c^*)=\\frac{1}{e\\lambda}}.\n$